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Displaying similar documents to “On certain Diophantine systems with infinitely many parametric solutions and applications”

Rank of elliptic curves associated to Brahmagupta quadrilaterals

Farzali Izadi, Foad Khoshnam, Arman Shamsi Zargar (2016)

Colloquium Mathematicae

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We construct a family of elliptic curves with six parameters, arising from a system of Diophantine equations, whose rank is at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals (p³,q³,r³,s³) not necessarily representing genuine geometric objects. It turns out that, as parameters of the curves, the integers p,q,r,s along with the extra integers u,v satisfy u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions. ...

Division-ample sets and the Diophantine problem for rings of integers

Gunther Cornelissen, Thanases Pheidas, Karim Zahidi (2005)

Journal de Théorie des Nombres de Bordeaux

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We prove that Hilbert’s Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called set of integers and of an elliptic curve of rank one over K ). We relate division-ample sets to arithmetic of abelian varieties.

Counting rational points near planar curves

Ayla Gafni (2014)

Acta Arithmetica

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We find an asymptotic formula for the number of rational points near planar curves. More precisely, if f:ℝ → ℝ is a sufficiently smooth function defined on the interval [η,ξ], then the number of rational points with denominator no larger than Q that lie within a δ-neighborhood of the graph of f is shown to be asymptotically equivalent to (ξ-η)δQ².

Parametric Solutions of the Diophantine Equation A² + nB⁴ = C³

Susil Kumar Jena (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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The Diophantine equation A² + nB⁴ = C³ has infinitely many integral solutions A, B, C for any fixed integer n. The case n = 0 is trivial. By using a new polynomial identity we generate these solutions, and then give conditions when the solutions are pairwise co-prime.